For questions about the quaternions: a noncommutative four dimensional division algebra over the real numbers.

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Does anyone know any resources for Quaternions for truly understanding them?

I've been studying Quaternions for a week, on my own. I've learned various facts about them but I still don't understand them. My goal is to understand rotation quaternions specifically. I don't want ...
0
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1answer
34 views

Need help interpreting an equation from an article (related to quaternions).

At this link, about half way down the page, there is an equation I don't understand http://physicsforgames.blogspot.com/2010/02/quaternions-why.html This is the equation. $$VV† = -x^2I^2 - y^2J^2 - ...
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3answers
291 views

How do you construct the quaternion and the multiplication rules, like Hamilton did?

So, I understand complex number multiplication, and how it represents $2D$ rotations. What I don't understand is, how you add two more imaginary numbers $j$ and $k$, and get $3D$ rotations. I believe ...
3
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1answer
168 views

The multiplication of 2D vectors produces what?

I am trying to learn about rotation quaternions, and in the process I am currently looking at 2D vector multiplication. To avoid confusion with other types of multiplication, this is the basic form I ...
3
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2answers
59 views

What type of division is possible in 1, 2, 4, and 8 but not the 3rd dimension?

In this article that talks about some history of hamilton http://plus.maths.org/content/curious-quaternions There is a snippet that says this: Multiplication is very sneaky. You can only set up ...
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1answer
41 views

How does one derive this rotation quaternion formula?

given an angle and an axis, the corresponding quaternion can be computed like this. $w = \cos( Angle/2)$ $x = \text{axis}.x * \sin( Angle/2 )$ $y = \text{axis}.y * \sin( Angle/2 )$ $z = ...
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0answers
147 views

How is the Quaternion multiplication derived?

Quaternion multiplication seems suspiciously similar to the cross product. How is it derived? Here is a description of the multiplication: Let $Q_1$ and $Q_2$ be two quaternions, which are defined, ...
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3answers
130 views

Quaternions: Difference(s) between $\mathbb{H}$ and $Q_8$

What is the difference between $\mathbb{H}$ and $Q_8$? Both are called quaternions.
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1answer
154 views

How do I calculate a position in front of a quaternion given the initial position and the quaternion?

Well I want to determine the position in front of, lets say 'an object'. 'An objects' has a position (pX,pY,pZ) and a quaternion rotation (qX,qY,qZ,qW) (where qW seems always to be ...
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1answer
153 views

How are these formulas for Quaternion -> Rotation Matrix related?

I'm trying to write a program to convert a quaternion to a rotation matrix. One source I found is: ...
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1answer
114 views

Quaternions, torque, and impulse.

In a physics simulation I have a solid ball of mass $m$ and moment of interia $M$ (which is a diagonal matrix with all entries equal to ${2\over5}mr^2=i$). Its instantaneous rotation is given by a ...
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2answers
507 views

half sine and half cosine quaternions

Something is a little bit unclear to me. In the image below you see that you need to divide the angle by a half. Acccording to wikipedia they say that this is so that I could rotate clockwise or ...
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4answers
940 views

Quaternions: why does ijk = -1 and ij=k and -ji=k

Currently i am studying quaternions. I do understand that i, j and k are imaginairy numbers. so $i^2 = j^2 =k^2 = -1$. But I could not understand this: ...
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2answers
273 views

Detail in the proof of the quaternion rotation identity

I am trying to understand the proof of the quaternion rotation identity illustrated in wikipedia ...
5
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2answers
186 views

$x^2+1=0$ uncountable many solutions [duplicate]

Possible Duplicate: Why are the solutions of polynomial equations so unconstrained over the quaternions? Coudl someone explain me the following: Why should $x^2+1=0$ have uncountable ...
2
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1answer
236 views

Matrix representation of the Quaternions?

Can anyone explain how why the matrix representation of the quaternions using real matrices is constructed as such?
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1answer
76 views

How can I break down a rotation of known amount around a known axis into two rotations of unknown amounts around known axes?

I have a vector that's been rotated a known amount about a known axis. I would like to break this rotation down into two separate rotations around known, linearly independent axes where the amounts I ...
4
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1answer
135 views

How to Perform Quaternion Multiplication

Everywhere that I've looked, it seems to be assumed that $i^{2} = j^{2} = k^{2} = - 1$, along with the other rules of quaternion multiplication. However - for my homework - I'm being asked to show ...
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1answer
95 views

inner product on matrices with quaternionic entries

let $\mathbb{H}$ be the quaternions and $Mat(n,\mathbb{H})$ be the vector space of $n\times n$ matrices over $\mathbb{H}$. Let $H(n,\mathbb{H}):=\{ A\in Mat(n,\mathbb{H}): \overline{A}^t=A \}$ be the ...
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1answer
456 views

3 axis gimbal controller and quaternions

this question has been probably asked in different forms but please bare with me: I'm building a three axis gimbal controller as part of my uni project. Besides the gimbal stabilization on each axis, ...
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2answers
470 views

Updating a quaternion orientation by a vector of euler angles

I'm trying to understand why this formula works to update an orientation with an angular velocity represented as a vector of rotations in $radians/{second}$. I understand that two quaternions ...
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1answer
161 views

how to extrapolate quaternions?

from http://answers.unity3d.com/questions/168779/extrapolating-quaternion-rotation.html ...
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1answer
77 views

Error in Weyl character formula computation.

I need someone with a keen eye for errors. I am trying to use the Weyl character formula for the symplectic group Sp$(4,\mathbb{C})$ on certain matrices coming from 2x2 quaternion matrices. Summing ...
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2answers
736 views

What do we lose passing from the reals to the complex numbers?

As normed division algebras, when we go from the complex numbers to the quaternions, we lose commutativity. Moving on to the octonions, we lose associativity. Is there some analogous property that we ...
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1answer
75 views

Sub division rings of division rings

Below $^\ast$ denotes "nonzero elements of". There is a problem in Jacobson's Basic Algebra 1, there is a problem to this effect: if $S$ is a subdivision ring of $\mathbb{H}$ such that $S^\ast$ is a ...
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1answer
148 views

Triple quaternion multiplication

I'm self learner and for some reason I can't wrap my head around quaternion multiplication. I just stumble upon one of equation in my text. Can anyone show step-by-step workout for below: $$ ...
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3answers
161 views

How can we make $\mathbb{R}^n$ into a multiplicative group?

Are there any sorts of multiplicative group structures we can put on set $\mathbb{R}^n$? By multiplicative, I mean that it should be compatible with the natural scalar multiplication on ...
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1answer
69 views

How $v=(-\sin\frac{\alpha}{2}\sin\frac{\beta}{2},\cos\frac{\alpha}{2}\sin\frac{\beta}{2},\sin\frac{\alpha}{2}\cos\frac{\beta}{2})$ is derived from…

In the book Quarternion and Rotation Sequences, I can't seem to work out how the final equation (colored in $\color{red}{red}$) is derived from the original equation (colored in $\color{blue}{blue}$). ...
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1answer
426 views

3D points rotation to quaternions

For the simplicity, we'll consider two 3D points, that moves one relatively to other, in time. Let's say: ...
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1answer
51 views

quaternion product distributivity

If you check the quaternion product derivation at wikipedia: http://en.wikipedia.org/wiki/Quaternion#Hamilton_product You can see that it is derived from a multiplication table between the ...
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1answer
348 views

The Join of Two Copies of $S^1$

So I know the fact that the join of $S^1$ and $S^1$ is homeomorphic to the 3-sphere, but I'm having trouble "seeing" this. I'd prefer something that appeals to geometric intuition, but more formal ...
5
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1answer
234 views

Why do Quaternions and octonions exist?

Ok so I have known about imaginary numbers for quite some time now. I also understand why we want them to exist (to have a solution for $x^2=-1$). I also remember reading that the complex numbers are ...
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2answers
94 views

Correspondence between rotation representations

I was wondering if there is a bijection between unit quaternions and other rotation representations such as vector of rotation, Euler angles or rotation matrices. It seems to me this is not the case ...
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1answer
393 views

Quaternion Differentiation

I have an application that tracks an image and estimates its position and orientation. The orientation is given by a quaternion, and it is modified by an angular velocity every frame. To predict the ...
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0answers
142 views

how to fine the quaternion between two vectors lies on two different Cartesian coordinate systems

I have two vectors lies on different cartesian coordinate systems. I want to find the quaternion between these two vectors. how I can do that?
4
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1answer
322 views

Uniform distributions on the space of rotations in 3D

I believe on moral grounds that the following three definitions are equivalent, and determine "the" uniform distribution on rotations in three dimensions. The Haar measure on $SO(3)$. The uniform ...
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1answer
241 views

how to get rotation component of quaternion form using 3d coordinates

I have a series of 3d coordinates distributed in a 3d space according to a root point. I can determine the x, y , z component using reducing the vectors. but I am not clear how to get rotation ...
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9answers
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Why is the complex number $z=a+bi$ equivalent to the matrix form $\left(\begin{smallmatrix}a &-b\\b&a\end{smallmatrix}\right)$ [duplicate]

Possible Duplicate: Relation of this antisymmetric matrix $r = \left(\begin{smallmatrix}0 &1\\-1&0\end{smallmatrix}\right)$ to $i$ On Wikipedia, it says that: Matrix ...
6
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1answer
228 views

How to identify $SL(2,\mathbb{C})/SU(2)$ and the hyperbolic 3-space?

I know that every coset representative $g\in SL(2,\mathbb{C})$ for $SL(2,\mathbb{C})/SU(2)$ can be chosen of the form $$ g = \left( \begin{array}{cc} \sqrt{t} & \frac{z}{\sqrt{t}}\\ 0 & ...
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1answer
361 views

Quaternions and Rotations

Two of the interesting achievements in Mathematics are Classification of platonic solids, and also classification of finite groups acting on the unit sphere in $\mathbb{R}^3$, and they are very nicely ...
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0answers
198 views

Quaternions in olympiad 3d geometry

It's known that we can use complex numbers to solve some 2d problems easier than synthetic methods. But, what do you think about using complex numbers in 3d geometry? I've found extend of complex ...
0
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1answer
86 views

Why is the square of quaternions not half of all axes angles expressed by this quaternion?

I want to devide a rotation, which is expressed as a quaternion. So I am doing it with Quaternion^POWER, where power is lower than 0. See my question before: here If I calculate following example ...
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2answers
140 views

Why does $i$ have infinitely many conjugates in $\mathbb{H}$?

Browsing this question: Why are the solutions of polynomial equations so unconstrained over the quaternions?, the pdf linked in the comments says that the infinitely many conjugates of $i$ in ...
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1answer
3k views

Compute Angle Between Quaternions (in Matlab)

I am working on a project where I have many quaternion attitude vectors, and I want to find the 'precision' of these quaternions with respect to each-other. Without being an expert in this type of ...
2
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1answer
186 views

Unit elements in Hurwitz quaternions

Hurwitz quaternions are defined as: $$H_u = \left\{a+bi+cj+dk\mid (a,b,c,d) \in\mathbb{Z}\mbox{ or }(a,b,c,d) \in\mathbb{Z}+\frac{1}{2}\right\}$$ (that is, all integer or half integer quaternions). ...
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1answer
125 views

Left and Right Vector bundles

I am reading a paper that starts talking about 'left vector bundles' and I'm having trouble figuring out what they mean. The specific setup is as follows: A quarternionic line bundle $L$ over ...
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3answers
527 views

How to get a part of a quaternion? e.g. get half of the rotation of a quaternion?

if I have a quaternion which describes an arbitrary rotation, how can I get for example only the half rotation or something like 30% of this rotation? Thanks in advance!
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1answer
332 views

Splitting of quaternion algebras

A rational (definite) quaternion algebra is an algebra of the form $$ \mathcal{K} = \mathbb{Q} + \mathbb{Q}\alpha + \mathbb{Q}\beta + \mathbb{Q}\alpha \beta $$ with $\alpha^2,\beta^2 \in ...
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0answers
337 views

How to convert Yaw, Pitch, Roll and Acceleration value to cartesian system?

I am having readings of Yaw, pitch, Roll, Rotation matrix, Quaternion and Acceleration. These reading are taken with frequency of 20 (per second). They are collected from the mobile device which is ...
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2answers
145 views

Why is any proper division subring of $\mathbb{H}$ contained in the center $Z(\mathbb{H})$?

Here is an idea I've been working on for self study. Suppose $S$ is a division subring of $\mathbb{H}$ (the quaternions, viewed as a subring of $M_2(\mathbb{C})$), which is stabilized by the maps ...